Analysing Mathematical Models of Intracellular Calcium Dynamics using Geometric Singular Perturbation Techniques

Abstract

Oscillations in free intracellular calcium (Ca2+) concentration are known to act as signals in almost all cell types, transmitting messages which control cellular processes including muscle contraction, cellular secretion and neuronal firing. Due to the universal nature of calcium oscillations, understanding the physiological mechanisms that underlie them is of great importance. A key feature of intracellular calcium dynamics that has been found experimentally is that some physiological processes occur much faster than others. This leads to models with variables evolving on very different time scales. In this thesis we survey a range of representative models of intracellular calcium dynamics, using geometric singular perturbation techniques with the aim of determining the usefulness of these techniques and what their limitations are. We find that the number of distinct time scales and the number of variables evolving on each time scale varies between models, but that in all cases there are at least two time scales, with at least two slower variables. Using geometric singular perturbation techniques we identify parameter regimes in which relaxation oscillations are seen and those where canard induced mixed mode oscillations are present. We find that in some cases these techniques are very useful and explain the observed dynamics well, but that the theory is limited in its ability to explain the dynamics when there are three or more distinct time scales in a model. It has been proposed that a simple experiment, whereby a pulse of inositol (1,4,5)- trisphosphate (IP3) is applied to a cell, can be used to distinguish between two competing mechanisms which lead to calcium oscillations [53]. However, detailed mathematical investigation of models has identified an anomalous delay in the pulse responses of some models, making interpretation of the experimental data difficult [14]. In this thesis we find that the response of models to a pulse of IP3 can be understood in part by using geometric singular perturbation techniques. Using recently developed theory for systems with three or more slow variables, we find that the anomalous delay can be due to the presence of folded nodes and their corresponding canard solutions or due to the presence of a curve of folded saddles. This delay due to a curve of folded saddles is a novel delay mechanism that can occur in systems with three or more slow variables. Importantly, we find that in some models the response to a pulse of IP3 is contrary to predictions for all bifurcation parameter values, which invalidates the proposed experimental protocol.